Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
You are right. Probably the observation that $\lambda_k/n$ converges to $\mu_k/(n+1)$ is only true for some special cases. The example I tried was $a_{ij} = \rho^{|i-j|}$, where $0 < \rho < 1$.
